Graphs with maximum connectivity index

Abstract

Let G be a graph and d v the degree (=number of first neighbors) of its vertex v. The connectivity index of G is χ=∑ ( d u d v ) −1/2, with the summation ranging over all pairs of adjacent vertices of G. In a previous paper (Comput. Chem. 23 (1999) 469), by applying a heuristic combinatorial optimization algorithm, the structure of chemical trees possessing extremal (maximum and minimum) values of χ was determined. It was demonstrated that the path P n is the n-vertex tree with maximum χ-value. We now offer an alternative approach to finding (molecular) graphs with maximum χ, from which the extremality of P n follows as a special case. By eliminating a flaw in the earlier proof, we demonstrate that there exist cases when χ does not provide a correct measure of branching: we find pairs of trees T, T′, such that T is more branched than T′, but χ( T)> χ( T′). The smallest such examples are trees with 36 vertices in which one vertex has degree 31.